Sine-triple-angle circle

In triangle geometry, the sine-triple-angle circle is one of a circle of the triangle.^[1]^[2] Let Template:Math and Template:Math points on Template:Mvar , a side of triangle Template:Mvar . And, define Template:Math and Template:Math similarly for Template:Mvar and Template:Mvar. If

$∠ A = ∠ A B_{1} C_{1} = A C_{2} B_{2},$

$∠ B = ∠ B C_{1} A_{1} = B A_{2} C_{2},$

and

$∠ C = ∠ C A_{1} B_{1} = C B_{2} A_{2},$

then Template:Math and Template:Math lie on a circle called the sine-triple-angle circle.^[3] At first, Tucker and Neuberg called the circle "cercle triplicateur".^[4]

Properties

$| A_{1} A_{2} | : | B_{1} B_{2} | : | C_{1} C_{2} | = \sin 3 A : \sin 3 B : \sin 3 C$ .^[5] This property is the reason why the circle called "sine-triple-angle circle". But, the number of circle which cuts three sides of triangle that satisfies the ratio are countless. The centers of these circles are on the hyperbola through the incenter, three excenters, and X(49) (see below for XTemplate:Sub).^[6]
The homothetic centers of Nine-point circle and the circle are the Kosnita point and the focus of Kiepert parabola.
The homothetic centers of circumcircle and the circle are X(184), the inverse of Jerabek center in Brocard circle, and X(1147).^[7]
Intersections of Polar of Template:Mvar and Template:Mvar with the circle and Template:Mvar and Template:Mvar are colinear.^[8]
The radius of sine-triple-angle circle is

$\frac{R}{| 1 + 8 \cos (A) \cos (B) \cos (C) |},$

The center of sine-triple-angle circle is a triangle center designated as X(49) in Encyclopedia of Triangle Centers.^[7]^[9] The trilinear coordinates of X(49) is

$\cos (3 A) : \cos (3 B) : \cos (3 C)$ .

$∠ A_{1} C_{1} A_{2} = (2 n - 1) A - (n - 1) π,$

$∠ B_{1} A_{1} B_{2} = (2 n - 1) B - (n - 1) π,$

and

$∠ C_{1} B_{1} C_{2} = (2 n - 1) C - (n - 1) π,$

then Template:Math and Template:Math are concyclic.^[8] Sine-triple-angle circle is the special case in n=2.

Also,

$| A_{1} A_{2} | : | B_{1} B_{2} | : | C_{1} C_{2} | = \sin (2 n - 1) A : \sin (2 n - 1) B : \sin (2 n - 1) C$ .